Gen. Physiol. Biophys. (1987), 6, 6 5 - 77
65
Kinetics of Non-Equilibrium Metabolism-Coupled Passive
Transport in Biosystems
Š. BALÁŽ, E. ŠTURDÍK and J. AUGUSTÍN
Department of Biochemical Technology, Slovak Technical University, Jánska 1, 82137 Bratislava,
Czechoslovakia
Abstract. Expressions for time course of solute concentration in an arbitrary
compartment of a biosystem were derived using simplifying assumptions of
unidirectional transport and first order metabolism kinetics. The coefficients of
the resulting exponential-summation function comprise, in addition to the
volumes and the connecting areas of individual compartments, the rate parameters of the processes mentioned. The equations presented were verified using
results obtained in drug potency testing.
Key words: Biokinetics — Passive transport — Hydrophobicity — Structureactivity relationship — Metabolism-coupled transport
Introduction
One of the problems concerning the kinetics of drug activity is the time-dependent concentration of the effector in the receptor region. Providing that only a
negligible fraction of the total drug amount added to a biosystem is bound to
the receptors, the distribution of the drug molecules can be treated separately
from their interactions with the receptor sites (Baláž et al. 1985). So far attempts
to solve this task have been made especially in the field of drug design, transport
through membrane lipid bilayers (Penniston et al. 1969; Kubinyi 1979; Cooper
et al. 1981; Baláž et al. 1984; Baláž and Šturdík 1984) and also the first order
elimination (Dearden and Townend 1978; Kubinyi 1979; Aarons et al. 1982;
Baláž and Šturdík 1985) being considered as major steps determining the fate
of drugs in the system. Differential equations derived to describe the above
processes had to be solved numerically since reversible transport was involved.
However, for sufficiently large biosystems and/or when the rates of transport
and metabolism are comparable, undirectional transport seems to be a good
Baláž et al.
66
representation of the in vivo situation. Thanks to this approximation the corres­
ponding differential equations are completely integrable. This communication is
aimed to present the explicit expressions for the solute concentration in arbi­
trary compartment, the cellular structure of biosystems being also considered.
Under the cellular structure periodical alterations of compartments with identi­
cal properties are meant. In this context, an eucaryotic cell may also have cellular
structure since it is subdivided by the endoplasmic reticulum and membranes of
subcellular organelles into a series of compartments.
Methods
An exact mathematical description of the solute distribution in biosystems would require the use of
the 2nd Fick law for each aqueous and lipidic phase of the system, taking into account their
inhomogeneity (interfacial diffusion layers more ordered than the bulks) as well as different solva­
tion of substances in individual phases (discontinuous concentration gradients at the interfaces).
Diffusion proceeds much more rapidly in the bulks than in the diffusion layers, the equilibrium being
attained within 0.1 s under the real in vivo conditions (Baláž et al. 1984; Baláž and Šturdík 1984).
To simplify the situation we shall assume instantaneous diffusion in the bulks which seems quite
plausible with respect to most experimental procedures in membrane research. In addition, the
practically homogeneous composition of the bulks allows to use a common description for the
reaction kinetics. There is quite a lot of reactants in real biosystems which are either present in
sufficiently high concentrations or buffered, so that their changes in the course of the reaction can
be neglected. Thus, the kinetics of metabolism in the /-th compartment is characterized by a global
rate constant k;.
N
*l=I*J«i
j =
(D
I
where ki is the rate constant for the;-th reaction, they-th reactant is present in concentration cľ N
is the total number of reactants in the /-th compartment.
Transport through the interfaces with the diffusion layers can be characterized (Penniston et al.
1969) by the transport rate parameters /(/, — direction from water to membrane, l2 — backwards).
Under the above assumptions, the kinetics of transport coupled with chemical reactions in an
TV-compartment system (Fig. 1) is described by
-i = Bc
(2)
where c and č are the vectors of concentrations in the individual compartments and their time
derivatives, respectively, the matrix B {N x N) has elements: b, , = AJJV, + k, (i = 1,2, ... N; K, is
the volume of the /-th phase; A, is the contact area between the /-th and the (/ + 1 )-th compartments,
/4N = 0; for the values of x see table l)and6, , , , = A{IJVU , (/ = 1, 2, ... N - 1). At time t = 0 the
solute (n moles) is present in the 1st compartment only. The equation (2) was solved by standard
methods (Benson 1960; Wolf et al. 1977).
The constants in empirical equations (10) and (12) (Table 3) were optimalized by non-linear
regression analysis (Fletcher and Powell 1963). The number of constants used to calculate statistical
parameters also involved the nonoptimalized values, the constant E from equation (12) was taken
once only.
Kinetics of Metabolism-Coupled Transport
67
Fig. 1. Schematical representation of the Ncompartment model system. V — volumes of
individual compartments, A — interfacial areas,
/ — transport rate parameters in direction water-lipid and vice versa (subscripts 1 and 2, respectively), k — reaction rate constants.
vN,
,k„
,K4
A,
A3 \
A4
Results
There are two types of solution to equation (2) depending on whether the
biosystem in question is cellular (i.e. composed of periodically alternating
compartments with identical properties) or not. For the latter case, the time
dependence of concentration (cm) in the w-th compartment is given by
m — 1
m
m
m
]
cm = nl\l\ \\A%X\ Vr I exp(-V)/n (K 1 = 1
1 = 1
J = l
ty
(3)
n = I
where
The values of x, y, z are specified in Table 1.
Table 1. Symbol explanations
m, i, N
n
x
y
z
even
odd
1
2
2
1
m/2
(m - l)/2
(m - 2)/2
(m - l)/2
When certain compartments have identical properties the corresponding
diagonal elements (i.e. eigenvalues in this case) of the matrix B are also equal
and, consequently, equation (2) is invalid. This phenomenon might not only be
observable in tissues of higher organisms but also in eucaryotic cells, which are
subdivided into a set of compartments by the endoplasmic reticulum and other
intracellular membranes. Although the shapes of these compartments are less
regular than those of the cells in tissues, identical properties may be quite
frequent. When the cellular periodicity starts with the M-th compartment
(which is obviously of lipidic nature and, consequently, M is even), the diagonal
elements AM to AN of the matrix B can be written as follows:
68
Baláž et al.
^M'2/
"ť ^M
" M
—
AM
+
2
'2/
" M
+ 2 + ^M + 2
+
+
••• ^*N + 1 - w '2/ MM + I - w +
^N + I - w
A M+ l'l/ *M + I + ^M + 1
= ... AN
—
=
(4)
^ M + 3 M/ ' M+3 + ^M +3
[ _ x /, / vN
+
1 _x +
KN
=
1 _x
+
(5)
The values of w, x are summarized in Table 1. Under the above conditions, the
time dependence of the concentration in the first (M + 1) compartments is given
by equation (3), and equation (6) holds for that of the (M + m)-th compart­
ment:
/M + I
C
M +m=
^ M +m\
2-i
y
a
u M+m^
2^flM+2i. M+ m'
' + ť
+
i =
+ e-' M + " I > M + 2i+1 . M + n /)
i=1
(6)
/
where
EM
+ m
= (n/VM
_ J/<M + 3>''2 + - » / < " + 3 >' 2 -»-»(>í M /K M )* + '" *
+2
M+I
MM
/ + 1
- W EI W / n )
+ 1 /^M + , )
i =
i
The coefficients a are given by the recurrent formulae:
«!,M + m =tf,.M+ m-l/(^M + 2-x - Aj); i = 1, 2, ... ( M - 1)
a
M+2i+4-x. M+m=
a
a
a
M + 2 i + 2-x. M + m - l / ( * +
M + m - I. M + m =
M + 2i + 2 - w. M + m =
a
Ui '
=
M + m - I . M + m - l/(^M + 2- x —
\aM +2i + 2 - w , M + m - I ~~ ( ' T
^ , 1> ••• ( j
-
U
^ M + 2 - w) i
U ^ M + 2i + 4 - w. M - n t J / Í ^ M + 2 - w
- ^ M + 2 - X ) ; ' = 0, 1, ... ( Z -
—
1)
M+ 1
fl
M + 2-x. M + m
=
—
2J
i= 1
a
>- M + m
i # M+ 2- x
The expression for concentration in an arbitrary compartment can thus suc­
cessively be derived starting with the solution to equation (2) for the first
compartment:
cl=(n/FI)e-4l«
p
p
and setting J J =--- 1 and £
i= m
(7)
= 0 for m > p.
i = m
To compare out results with biological reality we shall use data about effects
of xenobiotics on biosystems. Our considerations were based on the idea that
Kinetics of Metabolism-Coupled Transport
69
under appropriate conditions the biological activity of a compound may monitor the concentration in the receptor region (Baláž et al. 1985). To avoid the
difficult determinations of all the parameters /, k, V, A in individual compartments we shall convert them, in accordance with the current practice in the field
of drug design, into exactly measurable physicochemical properties, which can,
in turn, be confronted with both the bioactivity indices and our results. The
transport rate parameters have been shown to depend on the partition coefficient
/"(Zwolinski et al. 1949):
/, = a P / ( / J P + l )
(8)
/2 = «/(/?p+n
(9)
where a = DJhL, B = DLhA/DAhL, D is the d'ffusion coefficient in the aqueous
(A) and lipid (L) diffusion layers with and effective thickness h. Since values of
a, P for actual biosystems are not available we could use values obtained with
organic solvent-water model systems since there is a physical resemblance
between solvent-water and membrane-water interfaces. The values of a, fí in
model systems are independent of the solute structure even if the permeants are
not members of a congeneric series, if they are partially or completely ionized,
or form ion pairs (Kubinyi 1978; Van de Waterbeemd et al. 1981). The quantities a and f3 thus become properties of the model system characterizing the
quality of the diffusion layers on both sides of its interface. As for certain
congeneric series, the assumption of a constant rate for the metabolic processes
is quite plausible, the partition coefficient P can play the role of a "referent"
physicochemical property, and we can try to compare the shape of the experimental dependence of bioactivity indices on P with the respective relationship
resulting from our description of the solute disposition in biosystems. The time
development of such a relationship is shown in Fig. 2. The curves are bilinear
Fig. 2. The dependences of concentration c in
the 5th compartment on the partition coefficient
P after 3.2 (1), 10 (2), 32 (3), 100 (4) and 320
(5) hours. Model specification: K, = 1 dm3,
V2 = V, = 0.5 dm3, VA = V5 = 0.3 dm3; A, =
= A2...A5 = ldm 2 ; fc, = k2 = k4 = 0; ky =
= k5= 1 h"', /, and l2 obey equations (8), (9)
with a = 0.286dmh-'; yS = 0.245 (Kubinyi
1978), n = 1 mol dm' 3 .
Baláž et al.
70
(i.e. composed of two linear parts connected by a curved portion) in the early
stages of the distribution, and become distorted later in a characteristic manner
(Dearden and Townend 1978). They have two maxima separated by a minimum
and can be considered as consisting of four linear parts with curved joining
portions, the leftmost and rightmost linears retaining the slopes of the shortterm bilinear curves. The slopes are integer and their values are typical for
individual compartments (Fig. 3, Table 2). This fact may significantly promote
the elucidation of mechanisms of action of drugs since the shape of the bioactivity-lipophilicity profile can, under certain conditions, indicate the nature and
the number of the receptor compartment.
Fig. 3. The concentration-lipophilicity profiles
in the first 5 compartments after 3.2 hours. For
model specification see Fig. 2, the numbers of
the compartments are indicated at the corresponding lines.
Table 2. The integer values of the slopes in the linear parts for the bilinear concentration-lipophilicity profiles in the n-th compartment and the corresponding values of the constants A and B, in
equation (10) (N = 1).
Nature of the n-th
compartment
aqueous
lipidic
Slopes
in - l)/2
n/2
(1 - «)/2
(2 - «)/2
A
Bt
(n - l)/2
nil
(l-«)
(1-H)
Fig. 4 illustrates the influence of metabolism rate (constant within the series
studied) on the concentration — lipophilicity relationship. The shape of the
curves remains generally unchanged although some minor deviations are observable.
Kinetics of Metabolism-Coupled Transport
71
In order to compare quantitatively our results with the biological reality it
is indispensable to find and empirical formula describing both types of curves
with a sufficient accuracy. For this purpose equation (10) proved to be useful:
log(c/c0) = A log P + X 5,log(C,T +!) + &
(10)
where c0 is the initial concentration in the entry compartment, A, B, N are
integer and C[, D' non-integer constants. Their values depend on the parameters
of the distribution system (volumes of individual phases, connecting areas, a, R)
log P
Fig. 4. The dependences of concentration c in
the 5th compartment on the partition coefficient
P after 32 hours. For model specification see
Fig. 2., except: k, = k5 = 0.01 (1), 0.1 (2), 0.32
(3), 1 (4), 3.2 (5), 10 (6) and 32 (7)h"'.
from equations (8) and (9) and time. TV is 2 and 1 in the short-term bilinear
curves for the 1st compartment and for those with higher numbers, respectively,
and N = 3 for the curves with two maxima. A and B, determine the magnitude
of slopes in the linear parts, their values for the bilinear curves are given in Table 2. The constants C[ are coupled with the position of the curvatures. The
relation between the individual constants in equation (10) and the curve parameters can be seen in Fig. 5. Equation (10) was fitted to data presented in Figs.
2—4 by non-linear regression analysis (Fletcher and Powell 1963), the magnitudes of individual constants are given in Tables 3 and 4. The values of
statistical parameters indicate that an almost perfect fit was obtained.
To adapt equation (10) to biological reality the model partition coefficient P
has to be substituted by the membrane-water partition coefficient PM via equation (11) (Collander 1951)
PM = const PE
(11)
Baláž et al
72
This introduces an additional constant E into equation (10). Assuming a onestep and reversible drug-receptor interaction with the same affinity constant for
all the drugs in question, equations (10) and (11) can be rewritten for isotoxic
concentrations cx (Baláž et al. 1985):
log(l/c x ) = ,41ogP E + £fi,log(C,/> E +l) + Z)
i =
(12)
i
The empirical equation allows a comparison of dependences obtained in our
experiments (Figs. 2—4) with results of drug potency testing.
Fig. 5. The relation between the individual constants in equation (10) and the curve parameters. Expressions at the curve describe the
slopes in the corresponding linear parts.
Fig. 6. The neurotoxicity-hpophilicity profile of
homologous n-alcohols (Kubinyi 1979). The line
corresponds to empirical equation given in
Table 5.
Table 3. The constants in equation (10) with N = 1 describing the dependences given in Figs. 2 and
3. A and fi, were non-optimalized. Statistical parameters: 17 points in all cases (for log P = —3 to
5 with the step 0.5), the lowest values of the correlation coefficient and of the F-test were r = 0.998
and F = 298, the highest value of the standard deviation Í = 0.145.
Fig.
Curve
A
B,
1
2
3
3
3
3
1
2
2
3
4
5
2
2
1
1
2
2
-4
-4
-1
-2
-3
-4
C\
3.802 x
3.311 x
4.169 x
4.467 x
3.236 x
3.802 x
D
10"
10
10
10
10
10-
-1.080
-9.032
-1.387
-5.892
-7.447
-1.080
x 10 l
x 10"'
x 10 '
x 10"1
Table 4. Optimalized values of constants in equation (10) describing the dependences given in Figs. 2—4. Superscript a indicates non-optimalized
values. Statistical parameters: 17 points in all cases, the lowest r = 0.999 and F = 302, the highest Í = 0.131 (for symbols see text to Table 3).
Fig.
2
2
2
3
4
4
4
4
4
4
4
Curve
3
4
5
1
1
2
3
4
5
6
7
A°
2
2
2
0
2
2
2
2
2
2
2
s,
-3.790
-10.24
-10.26
-r
-5.281
-4.230
-3.731
-3.790
-3.645
-3.267
-3.186
B2
4.942
16.64
18.06
V
5.282
4.473
4.422
4.942
5.590
5.803
6.861
B,
—4.153
-7.402
-8.800
—
-3.021
-3.240
-3.695
-4.153
-4.955
-5.540
-6.682
C!
4.667
6.310
69.18
1.995
5.495
9.550
6.166
4.677
4.266
5.012
5.495
C\
0.309
0.562
0.398
0.032
0.389
0.324
0.275
0.309
0.295
0.316
0.144
D'
Q
3.890 x
1.622 x
5.754 x
—
2.884 x
3.020 x
3.311 x
3.890 x
4.169 x
3.981 x
3.311 x
2
10~
10-2
10"3
10~2
102
10 2
10"2
10 2
10"2
10 2
-0.945
-2.976
2.936
0.213
1.794
1.512
0.154
-0.945
-2.028
-2.888
-3.554
74
Baláž et al.
The first two examples were taken from an excellent paper by Kubinyi (1979).
In Fig. 6, neurotoxicity of homologous H-alcohols in rats (expressed in terms of
doses causing a certain degree of ataxia) was plotted against lipophilicity. The
same relation for hemolytic activity of a-monoglycerides is given in Fig. 7, curve
1 representing a 2 % ethanolic solution and curve 2 an aqueous solution. The
lines correspond to empirical equations summarized in Table 5. As follows from
the integer values of the coefficients A and B{ (also see Table 2), it can be inferred
on the basis of the model used that the receptor compartment is of aqueous
nature and it represent the third phase when counted from the site of application. Kubinyi (1979) suggested micelle formation as a possible explanation of a
decrease in activity of higher homologs in aqueous solution as compared with
those in ethanolic solution. In our context, however, it is the effect of the ethanol
on the erythrocyte membrane rather than a break-up of micelles that seems to
be a more plausible explanation for the observation. This hypothesis is especially supported by the fact that the coefficients A and Bx have the values 1 and
— 2, respectively and E changes in both cases (Table 5). Kubinyi fitted the data
with his bilinear equation allowing to adopt continuous values for the coefficients A and 5, with fixed E = 1. Our statistical parameters are comparable
with those of Kubinyi as for the correlation coefficient; standard deviation and
the Fischer test, however, have somewhat worse values. This is due to a higher
number of coefficients included in our empirical equation. They were all used to
calculate the statistical parameters neglecting the fact that some of them {A, fi;)
have fixed values and are not optimalized.
logP
Fig. 7. The dependence on lipophilicity of
hemolytic activity in a series of a-monoglycerides in 2 % ethanolic (curve 1, full circles) and
in aqueous solution (curve 2, open circles). The
lines correspond to equations summarized in
Table 5.
Table 5. Optimalized values of constants in equation (12) describing the bioactivity-hpophilicity data from literature given in Figs. 6—8. For
meaning of the statistical parameters, see text to Table 4.
Fig. Curve
6
7
7
8a
8a
8a
a
b
2
—
1
2
1
2
3
A
ŕ
ib
ib
—
—
—
Bt
B2
-2"
-2b
-2b
1.784
5.015
9.526
—
—
—
-1.534
-4.888
-8.852
c,
1.759
2.768
3.689
3.565
6.257
4.971
x
x
x
x
x
x
-2
10
10"
10"
10" 3
10 4
10"
c2
D
E
n
r
Í
F
—
—
—
6.274 x 1 0 "
1.800 x 10 "
2.131 x 1 0 "
1.606
1.457
1.079
7.339
6.475
5.757
0.904
0.864
1.054
1.017
1.125
1.087
10
8
7
8
8
8
0.998
0.998
0.997
0.981
0.931
0.909
0.109
0.056
0.096
0.204
0.691
1.066
21.9
68.1
30.4
3.25
0.82
0.61
ÍO -" is used instead of P, n being a substituent lipophilicity constant (Hansch and Leo 1979)
non-optimalized values
Baláž et al
76
In Fig. 8, the analgesic activity of N-4-substituted l-(2-arylethyl)-4-piperidinyl-N-phenylpropanamides in rats (c is ED50 in mol/kg) at various times (Van
Bever et al. 1976) was plotted against lipophilicity represented by the sum of
^•-constants for the changing substituents (Hansch and Leo 1979). Five derivatives with R, = CH 3 and R3 = H or COCH2CH3 (numbering according to the
original paper) were not taken for the correlation. The lines correspond to the
empirical equations given in Table 5. Although statistical evaluation gave not
very good results, the example is worth of being mentioned since it illustrates the
time development of the bioactivity — lipophilicity profile which agreed with
our model (also see Fig. 2).
u~~
24
Í2
~
Fig. 8. Analgesic activity of N-4-substituted l-(2-arylethyl)-4-piperidinyl-N-phenylpropanamides in rats,
2 (curve 1, full circles), 4 (curve 2, half-open circles)
and 6 hours (curve 3, open circles) after i.v. injection
(Van Bever et al. 1976) vs. lipophilicity expressed as
the sum of ^-constants of the changing substituents
(Hansch and Leo 1979).
The presented quantitative comparison of biological data with our model
suggest that equations (3) and (6) possibly describe the distribution kinetics of
low-molecular solutes in biosystems.
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Final version accepted March 7, 1986